Mathematical Tools of Quantum Mechanics
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Transcript of Mathematical Tools of Quantum Mechanics
Chapter 2
Mathematical Tools of Quantum Mechanics
Hilbert space
• Let’s recall for Cartesian 3D space:
• A vector is a set of 3 numbers, called components – it can be expanded in terms of three unit vectors (basis)
• The basis spans the vector space
• Inner (dot, scalar) product of 2 vectors is defined as:
• Length (norm) of a vector
zzyyxx AeAeAeA
zzyyxx BABABABA
AAA
Hilbert space
Hilbert space
• Hilbert space:
• Its elements are functions (vectors of Hilbert space)
• The space is linear: if φ and ψ belong to the space then φ + ψ, as well as aφ (a – constant) also belong to the space
David Hilbert(1862 – 1943)
Hilbert space
• Hilbert space:
• Inner (dot, scalar) product of 2 vectors is defined as:
• Length (norm) of a vector is related to the inner product as:
David Hilbert(1862 – 1943)
d)()(*)(),(
d)()(*
d2)(
Hilbert space
• Hilbert space:
• The space is complete, i.e. it contains all its limit points (we will see later)
• Example of a Hilbert space: L2, set of square-integrable functions defined on the whole interval
David Hilbert(1862 – 1943)
d)()(*
Wave function space
• Recall:
• Thus we should retain only such functions ψ that are well-defined everywhere, continuous, and infinitely differentiable
• Let us call such set of functions F
• F is a subspace of L2
• For two complex numbers λ1 and λ2 it can be shown that if
2.A
1),( trdP rdtrCtrdP 32),(),(
Fr )(1
Fr )(2
Frr )()( 2211
Scalar product
• In F the scalar product is defined as:
• Properties of the scalar product:
• φ and ψ are orthogonal if
• Norm is defined as
2.A.1
rdrr )()(*,
*,,
22112211 ,,,
,*,*, 22112211
0,
rdrrdrr 2)()()(*,
,
Scalar product
• Schwarz inequality
,,,
Karl Hermann Amandus Schwarz
(1843 – 1921)
2.A.1
Linear operators
• Linear operator A is defined as:
• Examples of linear operators:
• Parity operator:
• (Multiplication by) coordinate operator:
• Differentiation operator:
)(')( rrA
22112211 AAA
)()( rr
)()( rxrX
xrrDx
)()(
2.A.1
Linear operators
• Product of operators:
• In general:
• Commutator:
• Example:
)()( rBArAB
)(, rDX x
BAAB
BAABBA ,
)()( rXDXD xx
)(rxxx
x
)()( rxxx
rx
xxr
xrx
xrx
)()()(
)(r 1, xDX
2.A.1
Orthonormal bases
• A countable set of functions
• is called orthonormal if:
• It constitutes a basis if every function in F can be expanded in one and only one way:
• Recall for 3D vectors:
)(rui
ijji ruru )(),(
i
ii rucr )()(
2.A.2
,ju
iiij ucu ,
iiij ucu ,
iiji uuc ,
i
ijic jc rdrruuc jjj
)()(*,
ijji ee
iiiecC
ii eCc
Orthonormal bases
• For two functions
• a scalar product is:
• Recall for 3D vectors:
j
jji
ii rucrrubr )()(;)()(
,
2.A.2
i
ii cb *
i
iicbCB
jjj
iii ucub ,
jijiji uucb
,
,*
ji
ijji cb,
*
i
ii
ii ccc 2*,
i
ii cb *,
Orthonormal bases
• This means that
• Closure relation
i
ii rucr )()(
2.A.2
i
ii ruu )(,
i
ii rurdrru )(')'()'(*
')'()()'(* rdrruru
iii
')'()()'(*)( rdrrurur
iii
)'()()'(* rrrurui
ii
Orthonormal bases
• δ-function:
)()( afdxaxxf
Orthonormal bases
• A set of functions labelled by a continuous index α
• is called orthonormal if:
• It constitutes a basis if every function in F can be expanded in one and only one way:
)(rw
)'()(),( ' rwrw
drwcr )()()(
2.A.3
,w ')'(, ' dwcw ')'(, ' dwcw
',)'( ' dwwc '')'( dc )(c
rdrrwwc
)()(*,)(
Orthonormal bases
• For two functions
• a scalar product is: ')()'()(;)()()( ' drwcrdrwbr
,
2.A.3
dcb )()(*
')()'(,)()( ' drwcdrwb
')(),()'()(* ' ddrwrwcb
')'()'()(* ddcb
dcdcc 2)()()(*,
dcb )()(*,
Orthonormal bases
• This means that
• Closure relation
drwcr )()()(
2.A.3
drww )(,
drwrdrrw )(')'()'(*
')'()()'(* rdrdrwrw
')'()()'(*)( rdrdrwrwr
)'()(),'()()'(* rrrwrwdrwrw
)'()(),( ' rwrw
Orthonormal bases
• Useful relationship:
2.A.3
)(21 udkeiku
Examples of orthonormal bases
• Let us apply Fourier transform to function ψ(x):
• Using functions of plane waves
• we can write:
dpepxipx
)(
21)(
2.A.3
dxexp
ipx
)(
21)(
ipx
p exv21)(
dpxvpx p )()()(
dxxxvp p )()(*)(
drwcr )()()( rdrrwc )()(*)(
)(),( ' xvxv pp
dxe
ppxi '
21
)'( pp
Examples of orthonormal bases
• For two functions
• a scalar product is:
')()'()(;)()()( ' dpxvpxdpxvpx pp
,
2.A.3
dppp )()(*
')()'(,)()( ' dpxvpdpxvp pp
')(),()'()(* ' dpdpxvxvpp pp
')'()'()(* dpdppppp
dpp 2)(,
dppp )()(*,
Examples of orthonormal bases
• This means that
• Closure relation
')()'()( ' dpxvpx p
2.A.3
dpxvv pp )(,
dpxvdxxxv pp )(')'()'(*
')'()()'(* dxxdpxvxv pp
')'()()'(*)( dxxdpxvxvx pp
)'()(),'()()'(* xxxvxvdpxvxv pppp
)'()(),( ' ppxvxv pp
Examples of orthonormal bases
• Let us consider a set of functions:
• The set is orthonormal:
• Functions in F can be expanded:
)()( 00rrrr
rdrrrrrr rr
)'()()(),( 00'00
000 )()()( rdrrrr
2.A.3
,0r ')'()(, 00'00
rdrrrr
')()'(, 0'0 00rdrr rr
',)'( 0'0 00rdr rr
'')'( 0000 rdrrr )( 0r
rdrr rr
)(,)(000
)'( 00 rr
00 )()(0 rdrrr
Examples of orthonormal bases
• For two functions
• a scalar product is: ')'()()(;)()()( 00'00 00
rdrrrrdrrr rr
,
2.A.3
000 )()(* rdrr
')'()(,)()( 00'00 00rdrrrdrr rr
')(),()'()(* 00'00 00rdrdrrrr rr
')'()'()(* 000000 rdrdrrrr
)()()()(*, 02
000 rdrdrr
000 )()(*, rdrr
Examples of orthonormal bases
• This means that
• Closure relation
00 )()()(0
rdrrr r
2.A.3
0,)(00
rdr rr
0)(')'()'(00
rdrrdrr rr
')'()()'( 000rdrrdrr rr
')'()()'()( 000rdrrdrrr rr
)'()()'( 000rrrdrr rr
)'()(),( 00'00rrrr rr
State vectors and state space
• The same function ψ can be represented by a multiplicity of different sets of components, corresponding to the choice of a basis
• These sets characterize the state of the system as well as the wave function itself
• Moreover, the ψ function appears on the same footing as other sets of components
2.B.1
State vectors and state space
• Each state of the system is thus characterized by a state vector, belonging to state space of the system Er
• As F is a subspace of L2, Er is a subspace of the Hilbert space
2.B.1
Dirac notation
• Bracket = “bra” x “ket”
• < > = < | > = “< |” x “| >”
2.B.2
Paul Adrien Maurice Dirac(1902 – 1984)
Dirac notation
• We will be working in the Er space
• Any vector element of this space we will call a ket vector
• Notation:
• We associate kets with wave functions:
• F and Er are isomporphic
• r is an index labelling components
2.B.2
Paul Adrien Maurice Dirac(1902 – 1984)
rEFr )(
Dirac notation
• With each pair ok kets we associate their scalar product – a complex number
• We define a linear functional (not the same as a linear operator!) on kets as a linear operation associating a complex number with a ket:
• Such functionals form a vector space
• We will call it a dual space Er*
2.B.2
Paul Adrien Maurice Dirac(1902 – 1984)
rE
22112211
,
Dirac notation
• Any element of the dual space we will call a bra vector
• Ket | φ > enables us to define a linear functional that associates (linearly) with each ket | ψ > a complex number equal to the scalar product:
• For every ket in Er there is a bra in Er*
2.B.2
Paul Adrien Maurice Dirac(1902 – 1984)
,
Dirac notation
• Some properties:
2.B.2
Paul Adrien Maurice Dirac(1902 – 1984)
2211,
2211
,2211
2211 **
,*,* 2211
*
2211 ,,
, ,* *
*
Linear operators
• Linear operator A is defined as:
• Product of operators:
• In general:
• Commutator:
• Matrix element of operator A:
' A
22112211 AAA
2.B.3
BAAB
BAAB
BAABBA ,
A
Linear operators
• Example:
• What is ?
• It is an operator – it converts one ket into another
2.B.3
,
Linear operators
• Example:
• Let us assume that
• Projector operator
• It projects one ket onto another
2.B.3
P
1
PPP 2
P
Linear operators
• Example:
• Let us assume that
• These kets span space Eq, a subspace of E
• Subspace projector operator
• It projects a ket onto a subspace of kets
2.B.3
qP
q
iii
1
q
iii
1
q
iiiqP
1
ijji
qqq PPP 2
q
jjj
q
iii
11
q
jijjii
1,
q
iii
1
qP
qji ,...,2,1,
q
jijiji
1,
Linear operators
• Recall matrix element of a linear operator A:
• Since a scalar product depends linearly on the ket, the matrix element depends linearly on the ket
• Thus for a given bra and a given operator we can associate a number that will depend linearly on the ket
• So there is a new linear functional on the kets in space E, i.e., a bra in space of E*, which we will denote
• Therefore
2.B.4
A
A
AA A
Linear operators
• Operator A associates with a given bra a new bra
• Let’s show that this correspondence is linear
2.B.4
' A
2211 2211
AA AA 2211
AA 2211
AA 2211 AA 2211 ... DEQ
Charles Hermite(1822 – 1901)
Linear operators
• For each ket there is a bra associated with it
• Hermitian conjugate (adjoint) operator:
• This operator is linear (can be shown)
2.B.4
''
A'
†' A
*'' *† AA
Charles Hermite(1822 – 1901)
Linear operators
• Some properties:
2.B.4
†† * AA AA ††
†A
AB
††† BABA
A B
†B †AB
†A ††AB ††† ABAB
Charles Hermite(1822 – 1901)
Hermitian conjugation
• To obtain Hermitian conjugation of an expression:
• Replace constants with their complex conjugates
• Replace operators with their Hermitian conjugates
• Replace kets with bras
• Replace bras with kets
• Reverse order of factors
2.B.4
* †AA
†AA
Charles Hermite(1822 – 1901)
Hermitian operators
• For a Hermitian operator:
• Hermitian operators play a fundamental role in quantum mechanics (we’ll see later)
• E.g., projector operator is Hermitian:
• If:
2.B.4
0, BA
†AA
††† ABAB
* AA
P †† P
AA † BB †
BA AB ABAB †
Representations in state space
• In a certain basis, vectors and operators are represented by numbers (components and matrix elements)
• Thus vector calculus becomes matrix calculus
• A choice of a specific representation is dictated by the simplicity of calculations
• We will rewrite expressions obtained above for orthonormal bases using Dirac notation
2.C.1
Orthonormal bases
• A countable set of kets
• is called orthonormal if:
• It constitutes a basis if every vector in E can be expanded in one and only one way:
iu
ijji uu
i
ii uc
ju i
iji uuc i
ijic jc
jj uc
2.C.2
Orthonormal bases
• Closure relation
• 1 – identity operator
i
ii uc
2.C.2
i
ii uu
ii
i uu
iii uu
iii uu
1̂}{ i
iiu uuPi
1̂
Orthonormal bases
• For two kets
• a scalar product is:
i
iii
ii ucub ;
2.C.3
i
ii cb *
1̂
i
ii uu
i
ii
ii ccc 2*i
ii cb *
iiii ucub ;
iii uu
Orthonormal bases
• A set of kets labelled by a continuous index α
• is called orthonormal if:
• It constitutes a basis if every vector in E can be expanded in one and only one way:
w
)'(' ww
dwc )(
2.C.2
w ')'( ' dwcw ')'( ' dwwc
'')'( dc )(c
wc )(
Orthonormal bases
• Closure relation
• 1 – identity operator
dww
2.C.2
dwc )(
dww dww
dww
1̂}{ dwwPw
1̂
Orthonormal bases
• For two kets
• a scalar product is:
dwcdwb )(;)(
2.C.3
dcb )()(*
1̂
dww
dc 2)( dcb )()(*
wcwb )(;)(
dww
Representation of kets and bras
• In a certain basis, a ket is represented by its components
• These components could be arranged as a column-vector:
...
...2
1
iu
uu
2.C.3
...
...
w
Representation of kets and bras
• In a certain basis, a bra is also represented by its components
• These components could be arranged as a row-vector:
......21 iuuu
2.C.3
...... w
Representation of operators
• In a certain basis, an operator is represented by matrix components:
...............
......
...............
......
......
21
22221
11211
ijii
j
j
AAA
AAAAAA
2.C.4
'............)',(............
A
jiij uAuA ')',( wAwA
jiji uBAuuABu 1̂
jk
kki uBuuAu
kjkki uBuuAu
Representation of operators2.C.4
jjji uuAu
A'
'' ii uc Aui 1̂Aui
j
jji uuAu j
jijcA
...
...
...............
......
...............
......
......
...'...''
2
1
21
22221
11211
2
1
iijii
j
j
i c
cc
AAA
AAAAAA
c
cc
Representation of operators2.C.4
''' dwwAw
')(' wc Aw 1̂Aw
''' dwwAw
')'()',( dcA
A'
Representation of operators2.C.4
...
...
...............
......
...............
......
......
...*...**2
1
21
22221
11211
121
iijii
j
j
c
cc
AAA
AAAAAA
bbb
A
iiii ucub ;
Representation of operators2.C.4
...*...**
...
... 21
2
1
j
i
cccc
cc
ii uc
...............
...*...**
...............
...*...**
...*...**
21
22212
12111
jiii
j
j
cccccc
cccccccccccc
Representation of operators
• For Hermitian operators:
• Diagonal elements of Hermitian operators are always real
2.C.4
jiij uAuA ††
'†† )',( wAwA
*ij uAu *jiA
*' wAw ),'(* A
*jiij AA *),'()',( AA
*iiii AA *),(),( AA
1SSSS ††
Change of representations2.C.5
i
iik uut
iiik uutkt 1̂kt
i
iki uS †
k
kki ttu
kkki ttuiu 1̂iu
k
kik tS
i
kii tuuki
ii tuu
kt kt1̂
i
iki Su
Change of representations2.C.5
ji
ljjiik tuuAuut,
lj
jji
iik tuuAuut
lk tAt
ji
jiijki SAS,
†
lk
jllkki uttAttu,
jl
llk
kki uttAttu
ji uAu
lk
ijklik SAS,
†
Eigenvalue equations
• A ket is called an eigenvector of a linear operator if:
• This is called an eigenvalue equation for an operator
• This equation has solutions only when λ takes certain values - eigenvalues
• If:
• then:
2.D.1
A
A
*† A
Eigenvalue equations
• The eigenvalue is called nondegenerate (simple) if the corresponding eigenvector is unique to within a constant
• The eigenvalue is called degenerate if there are at least two linearly independent kets corresponding to this eigenvalue
• The number of linearly independent eigenvectors corresponding to a certain eigenvalue is called a degree of degeneracy
2.D.1
Eigenvalue equations
• If for a certain eigenvalue λ the degree of degeneracy is g:
• then every eigenvector of the form
• is an eigenvector of the operator A corresponding to the eigenvalue λ for any ci:
• The set of linearly independent eigenvectors corresponding to a certain eigenvalue comprises a g-dimensional vector space called an eigensubspace
2.D.1
i
iicAA
i
iiAc
i
iic
i
iic
i
iic
giA ii ,...2,1;
Eigenvalue equations
• Let us assume that the basis is finite-dimensional, with dimensionality N
• This is a system of N linear homogenous equations for N coefficients cj
• Condition for a non-trivial solution:
2.D.1
ii uAu A
ij
jji uuuAu i
jjji uuuAu i
jjij ccA
0j
jijij cA
0 1A
Eigenvalue equations
• This equation is called the characteristic equation
• This is an Nth order equation in and it has N roots – the eigenvalues of the operator
• Condition for a non-trivial solution:
2.D.1
0 1A
0
...............
...
...
21
22221
11211
NNNN
N
N
AAA
AAAAAA
Eigenvalue equations
• Let us select λ0 as one of the eigenvalues
• If λ0 is a simple root of the characteristic equation, then we have a system of N – 1 independent equations for coefficients cj
• From linear algebra: the solution of this system (for one of the coefficients fixed) is
2.D.1
00 1A 00 j
jijij cA
1; 011
0 cc jj
j
jj uc0 j
jj uc10
jjj uc 0
1
Eigenvalue equations
• Let us select λ0 as one of the eigenvalues
• If λ0 is a multiple (degenrate) root of the characteristic equation, then we have less than N – 1 independent equations for coefficients cj
• E.g., if we have N – 1 independent equations then (from linear algebra) the solution of this system is
2.D.1
00 1A 00 j
jijij cA
0;1; 01
02
02
012
01
0 ccc jjj
j
jjj
jj ucuc 02
010
Eigenproblems for Hermitian operators
• For:
• Therefore λ is a real number
• Also:
• If:
• Then:
• But:
2.D.2
A†AA
A †* AA A
0Im A 0Im
A
A A
A A
A A 0
Observables
• Consider a Hermitian operator A whose eigenvalues form a discrete spectrum
• The degree of degeneracy of a given eigenvalue an will be labelled as gn
• In the eigensubspace En we consider gn linearly independent kets:
• If
• Then
2.D.2
,...2,1; nan
ninn
in giaA ,...,2,1;
'nn aa
0' jn
in
Observables
• Inside each eigensubspace
• Therefore:
• If all these eigenkets form a basis in the state space, then operator A is called an observable
2.D.2
ijjn
in
1̂1
n
g
i
in
in
n
'' nnijjn
in
Observables
• For an eigensubspace projector
• These relations could be generalized for the case of continuous bases
• E.g., a projector is an observable
2.D.2
ng
i
in
innP
1
n
g
i
in
in
n
AAA1
1̂
n
g
i
in
inn
n
a1
n
nnPa
n
nnPaA
P 1
Observables
• If
• Then
• If a is non-degenerate then
• so this ket is also an eigenvector of B
• If a is degenerate then
• Thereby, if A and B commute, each eigensubspace of A is globally invariant (stable) under the action of B
2.D.3
0, BA aA
aBBA aBAB
BaBA
B
aEB
Observables
• If
• Then
• If two operators commute, there is an orthonormal basis with eigenvectors common to both operators
2.D.3
0, BA
111 aA
21121 BaAB
222 aA 21 aa
021 B
21221 BaBA
21212121 BaaBAAB
Observables
• A set of observables, commuting by pairs, is called a complete set of commuting observables (CSCO) if there exists a unique orthonormal basis of common eigenvectors
• If all the eigenvalues of a certain operator are non-degenerate, this operator constitutes CSCO by itself
• If one ore more eigenvalues of a certain operator are degenerate, there is no unique orthonormal basis of eigenvectors
• Then at least one more operator commuting with the first one is used to construct a unique orthonormal basis of common eigenvectors, an thus a CSCO
2.D.3
Examples of representations
• Let us consider a set of functions:
• The set is orthonormal:
• Kets can be expanded:
00 )()(0
rrrrr
rdrrrrrr
)'()(' 0000
2.E.1
)'( 00 rr
000 )( rdrr
0r '')'( 0000 rdrrr
'')'( 0000 rdrrr
'')'( 0000 rdrrr
)( 0r
00 )( rr
Examples of representations
• Closure relation
000 rdrr
2.E.1
000 )( rdrr
000 rdrr
000 rdrr
000 rdrr
1̂000}{ 0 rdrrP r
1̂
Examples of representations
• For two kets
• a scalar product is:
000000 )(;)( rdrrrdrr
2.E.1
000 )()(* rdrr
1̂
000 rdrr
02
0 )( rdr
000 )()(* rdrr
0000 )(;)( rrrr
000 rdrr
Examples of representations
• Let us consider a set of functions:
• The set is orthonormal:
• Kets can be expanded:
03
0
02
1)( pervrpi
p
rdepp
ppri 00 '
300 21'
2.E.1
)'( 00 pp
000 )( pdpp
0p '')'( 0000 pdppp
'')'( 0000 pdppp
'')'( 0000 pdppp
)( 0p
00 )( pp
Examples of representations
• Closure relation
000 pdpp
2.E.1
000 )( pdpp
000 pdpp
000 pdpp
000 pdpp
1̂000}{ 0 pdppP p
1̂
Examples of representations
• For two kets
• a scalar product is:
000000 )(;)( pdpppdpp
2.E.1
000 )()(* pdpp
1̂
000 pdpp
02
0 )( pdp 000 )()(* pdpp
0000 )(;)( pppp
000 pdpp
Change of representations
• Recall:
• Choosing
• we obtain:
2.E.1
00 )( rr
rpi
p ervp
0
0 302
1)(
00
3002
1 rpi
epr
rpi
epr
32
1
pdppr
pdpprr 1̂r
00 )( pp
rdrrp rdrrpp
1̂p
pdeprpi
)(2 2/3
R and P operators
• For
• we obtain:
• where
• Similarly
• “Vector” operator R:
2.E.2
),,()( zyxrr
X'
),,(')('' zyxrr
),,(),,(' zyxxzyx rxr
' rxXr
ryYr
rzZr
ZkYjXiR ˆˆˆ
XX 1̂
rdXrr
Xrdrr
rdrxr rdrxr )()(*
R and P operators
• “Vector” operator P:
• Then:
2.E.2
ppPp xx
ppPp yy
ppPp zz
zyx PkPjPiP ˆˆˆ
xx PrPr 1̂
pdPppr x
xPpdppr
pdpppr x
pdppe x
rpi
)(2
13
x
ri
)(
R and P operators
• Analogously:
• Then:
2.E.2
xr
iPr x
)(
yr
iPr y
)( zr
iPr z
)(
)(ri
Pr
ri
xx PP 1̂
rdPrr x
xPrdrr
rdxr
ir
)()(*
R and P operators
• Calculating a commutator:
• Similarly:
2.E.2
XPXPrPXr xxx ,
Xrxi
Prx x
XPrXPr xx
rxxi
rxi
x
xxr
ir
xixr
xix
ri
iPX x ,
0],[],[ jiji PPRR
ijji iPR ],[
R and P operators
• Calculating a matrix element:
• Similarly:
• Position and momentum operators are Hermitian
2.E.2
rdrxrX )()(*
* X *)()(* rdrxr
rdrxi
rPx
)()(*
dxr
xrrrdydz
ix
x)(*)()()(*
dxrxi
rdydz )()(*
dxr
xirdydz )(*)(
* xP
R and P operators
• Calculating a matrix element:
• Thus:
• Similarly:
• Since |r > and |p > constitute complete bases, therefore operators R and P are observables
• Sets of operators {X,Y,Z} as well as {Px,Py,Pz} comprise a CSCO each, however, separate operators don’t, since they are degenerate (in other directions)
2.E.2
00 rrxrXr )( 00 rrx
)( 0rrx
00 rrx
00 rxrX
rzrZ
ryrY
rxrX
pppP
pppP
pppP
zz
yy
xx
Tensor products of state spaces
• Spaces of square-integrable functions in 1D, 2D, and 3D are not the same (e.g., Er and Ex are different)
• How are those spaces related?
• In general, if there are two or more mutually isolated subsystems of a certain system, each of which has its own space, what is the space of the entire system?
• Such questions are resolved via introduction of tensor products of spaces
2.F.1
Tensor products of state spaces
• Let there be two spaces E1 and E2 with dimensions N1 and N2
• Tensor product of E1 and E2 is a vector field E with the following properties:
• Notation:
• If vectors belonging to E1 and E2 are
• Then vectors belonging to E are
• Tensor product is linear:
2.F.2
21 EEE
)2(;)1(
)2()1(
)2()1()2()1()2()1(
Tensor products of state spaces
• Let there be two spaces E1 and E2 with dimensions N1 and N2
• Tensor product of E1 and E2 is a vector field E with the following properties:
• Tensor product is distributive:
• Tensor product of bases is a basis
2.F.2
)2()1()2()1(
)2()2()1(
21
21
21 )2(;)1( EvEu ji 21)2()1( EEEvu ji
Tensor products of state spaces
• If:
• Then:
• Components of a tensor product of two vectors are products of the components
• Not all the vectors in E can be represented as tensor products of vectors from E1 and E2:
2.F.2
j
jji
ii vbua )2(;)1(
ji
jiji vuba,
)2()1(
ji
jiji vuc,
jiji bac
Tensor products of state spaces
• Scalar product:
• For orthonormal bases:
• Tensor product of operators:
• Projector:
2.F.2
)2()2()1()1()2()1()2()1( BABA
)2()2()1()1()2()1()2()1(
jliklkji vuvu )2()1()2()1(
)2()2()1()1(
)2()1()2()1(
Tensor products of state spaces
• If:
• Then:
• And:
• Eigenvectors of A(1) + B(2) are tensor products of eigenvectors of A(1) and eigenvectors of B(2)
• If there is one CSCO in E1 and another CSCO in E2 their tensor product is a CSCO in E
2.F.3
)2()1()2()1()2( nmnnm bB
)2()2()2(;)1()1()1( nnnmmm bBaA
)2()1()2()1()1( nmmnm aA
)2()1(
)2()1()2()1(
nmnm
nm
ba
BA
Tensor products of state spaces
• If the problem is strictly 1D (e.g. x-dependent), then the state space is Ex
• In the x-representation the basis kets:
• Similarly we can consider Ey and Ez:
• Introducing
• we get
2.F.4
xxxxxx x )();()( 00 0
zyxxyz EEEE
yyyyyy y )();()( 00 0
zzzzzz z )();()( 00 0
zyxzyx ,,
Tensor products of state spaces
• Exyz is the state space of a 3D particle
• A ket in 3D can be represented:
• In general, ψ cannot be factorized
• Operator X in Ex is a CSCO by itself, but in Exyz its eigenvalues would be infinitely degenerate, because Ey and Ez in are infinitely-dimensional
• On the other hand, the {X,Y,Z} set is a CSCO
2.F.4
)()()()( 00000 zzyyxxrrrr
zyxzyx
zyxzyxdxdydz
,,),,(
,,),,(
States of a two-particle system
• For two particles the state space is:
• and the basis is:
• A ket can be represented:
• If:
• Then:
• In this case there is no correlation between the particles
2.F.4
2121, rrrr
2121 rrrr EEE
2121
212121
,),(
,),(
rrrr
rrrrrdrd
21
2121 ,),( rrrr 2211 rr
)()( 2211 rr